A dynamical approach to Vinogradov's theorem for ordered groups

A foundational result of Vinogradov asserts that the classes of ordered groups are closed under free products: free products of left-orderable groups are left-orderable, and the same holds for bi-ordered and circularly ordered groups.

From a dynamical perspective, a countable group is left-orderable if and only if it admits a faithful orientation-preserving action on the real line by homeomorphisms; similarly, a group is circularly ordered if and only if it admits a faithful action on the circle. These dynamical characterizations lead to a simple proof of Vinogradov’s theorem in the left- and circularly ordered cases: given actions of two groups on the line (the circle), one can naturally combine them to obtain an action of their free product, and faithfulness can be ensured by conjugating one of the factors by a suitable generic homeomorphism.

Countable bi-ordered groups can be characterized by admitting almost free actions on the real line, meaning that for every group element g, one has either g(x)≥x or g(x)≤x for all x. This property is typically not preserved under free products, which explains why Vinogradov’s theorem in the bi-ordered setting is significantly more complicated.

I will introduce an alternative dynamical construction for bi-ordered groups and use it to give a new proof of Vinogradov’s theorem.